I'd like it to move nearby vertices slightly when I edit a vertex
Use proportional editing.

 
(or press the O to enable and Alt+O to disable) and use the mouse wheel to determine the radius of the tool.




  Proportional Edit is a way of transforming selected elements (such as vertices) while having that transformation affect other nearby elements. For example, having the movement of a single vertex cause the movement of unselected vertices within a given range. Unselected vertices that are closer to the selected vertex will move more than those farther from it (i.e. they will move proportionally relative to the location of the selected element). Since Proportional Editing affects the nearby geometry, it is very useful when you need to smoothly deform the surface of a dense mesh.


Read the manual here: https://docs.blender.org/manual/en/latest/scene_layout/object/editing/transform/control/proportional_edit.html?highlight=proportional%20edit